Question

Can two electrons have the same quantum numbers?

Answer 1

No, two electrons can't have the same quantum number, this is explained as following:

There are four quantum numbers to describe the spatial position of electron in an atom.

$\mathrm{n}=$Principal Quantum number,

$\mathrm{n}=1,2,3,-----$

$\mathrm{l}=$azimuthal quantum number

$\mathrm{l}=0,1,2,3------$

$\mathrm{m}=$magnetic Quantum number

$\mathrm{m}=-\mathrm{l}$to $+\mathrm{l}$

$\mathrm{s}=$spin quantum number

$\mathrm{s}=-1/2,+1/2$

1. The Pauli Exclusion Theory states that "In an atom or molecule no two electrons have the same four electronic quantum numbers''.
2. The reason behind this is if one electron occupies the same set of quantum numbers as another electron, including the equivalent magnetic moment and direction of spin, its wave function turns out to be exactly the opposite of the other one.
3. The further reason is due to the hypothesis that electrons are fermions and thus their wave function is antisymmetric. The antisymmetric nature of wave function cancel each other to zero, which would violate the conservation of energy.

Eg;- Let us calculate all quantum numbers for $2{\mathrm{p}}^3$ and $2{\mathrm{p}}^4$ in Oxygen atom.

Electronic configuration of Oxygen atom: $1{\mathrm{s}}^{2}2{\mathrm{s}}^{2}2{\mathrm{p}}^4$

For $2{\mathrm{p}}^3$ , $$\begin{gathered} \mathrm{n}=2 \\ \mathrm{l}=1 \\ \mathrm{m}=-1,0,+1 \\ \mathrm{s}=+1/2 \end{gathered}$$

For $2{\mathrm{p}}^4$, $$\begin{gathered} \mathrm{n}=2 \\ \mathrm{l}=1 \\ \mathrm{m}=-1,0,+1 \\ \mathrm{s}=-1/2 \end{gathered}$$

The value of $\mathrm{n},\mathrm{l},\mathrm{m}$ is same for $2{\mathrm{p}}^3$ and $2{\mathrm{p}}^4$ in Oxygen atom. However the value of $\mathrm{s}$ is different for both configuration.

The above points and detailed explanation through examples clearly explains the different values of quantum number.